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Applications of Haar Wavelet-Finite Difference Hybrid Method and Its Convergence for Hyperbolic Nonlinear Schr ö dinger Equation with Energy and Mass Conversion

Author

Listed:
  • Xuan Liu

    (Department of Mathematics, Hanshan Normal University, Chaozhou 515041, China)

  • Muhammad Ahsan

    (Department of Mathematics, University of Swabi, Swabi 23200, Pakistan)

  • Masood Ahmad

    (Department of Basic Sciences, University of Engineering and Technology Peshawar, Peshawar 25000, Pakistan)

  • Muhammad Nisar

    (Department of Mathematics and Statistics, Macquarie University, Sydney, NSW 2109, Australia
    Department of Mathematics, FATA University, Darra Adam Khel 26100, Pakistan)

  • Xiaoling Liu

    (Department of Mathematics, Hanshan Normal University, Chaozhou 515041, China)

  • Imtiaz Ahmad

    (Department of Mathematics, University of Swabi, Swabi 23200, Pakistan)

  • Hijaz Ahmad

    (Department of Computer Engineering, Biruni University, Istanbul 34025, Turkey
    Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, 00186 Roma, Italy)

Abstract

This article is concerned with the numerical solution of nonlinear hyperbolic Schr o ¨ dinger equations (NHSEs) via an efficient Haar wavelet collocation method (HWCM). The time derivative is approximated in the governing equations by the central difference scheme, while the space derivatives are replaced by finite Haar series, which transform it to full algebraic form. The experimental rate of convergence follows the theoretical statements of convergence and the conservation laws of energy and mass are also presented, which strengthens the proposed method to be convergent and conservative. The Haar wavelets based on numerical results for solitary wave shape of | φ | are discussed in detail. The proposed approach provides a fast convergent approximation to the NHSEs. The reliability and efficiency of the method are illustrated by computing the maximum error norm and the experimental rate of convergence for different problems. Comparisons are performed with various existing methods in recent literature and better performance of the proposed method is shown in various tables and figures.

Suggested Citation

  • Xuan Liu & Muhammad Ahsan & Masood Ahmad & Muhammad Nisar & Xiaoling Liu & Imtiaz Ahmad & Hijaz Ahmad, 2021. "Applications of Haar Wavelet-Finite Difference Hybrid Method and Its Convergence for Hyperbolic Nonlinear Schr ö dinger Equation with Energy and Mass Conversion," Energies, MDPI, vol. 14(23), pages 1-17, November.
    • Handle: RePEc:gam:jeners:v:14:y:2021:i:23:p:7831-:d:685428
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    References listed on IDEAS

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    1. Ahsan, Muhammad & Ahmad, Imtiaz & Ahmad, Masood & Hussian, Iltaf, 2019. "A numerical Haar wavelet-finite difference hybrid method for linear and non-linear Schrödinger equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 165(C), pages 13-25.
    2. Hsiao, Chun-Hui & Wang, Wen-June, 2001. "Haar wavelet approach to nonlinear stiff systems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 57(6), pages 347-353.
    3. Saeed, Umer & ur Rehman, Mujeeb, 2015. "Haar wavelet Picard method for fractional nonlinear partial differential equations," Applied Mathematics and Computation, Elsevier, vol. 264(C), pages 310-322.
    4. Nazir, Shah & Shahzad, Sara & Wirza, Rahmita & Amin, Rohul & Ahsan, Muhammad & Mukhtar, Neelam & García-Magariño, Iván & Lloret, Jaime, 2019. "Birthmark based identification of software piracy using Haar wavelet," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 166(C), pages 144-154.
    5. Pervaiz, Nosheen & Aziz, Imran, 2020. "Haar wavelet approximation for the solution of cubic nonlinear Schrodinger equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 545(C).
    6. Hsiao, C.H., 2004. "Haar wavelet approach to linear stiff systems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 64(5), pages 561-567.
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    Cited by:

    1. Ahsan, Muhammad & Bohner, Martin & Ullah, Aizaz & Khan, Amir Ali & Ahmad, Sheraz, 2023. "A Haar wavelet multi-resolution collocation method for singularly perturbed differential equations with integral boundary conditions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 204(C), pages 166-180.
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